1. Converse Theorems

2. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.

3. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
A B

4. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
A B

5. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
 in a semicircle  90 

A B

6. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
 in a semicircle  90 

A B
(2) If an interval AB subtends the same angle at two points P and Q on
the same side of AB, then A,B,P,Q are concyclic.

7. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
 in a semicircle  90 

A B
(2) If an interval AB subtends the same angle at two points P and Q on
the same side of AB, then A,B,P,Q are concyclic.
P Q
 
A B

8. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
 in a semicircle  90 

A B
(2) If an interval AB subtends the same angle at two points P and Q on
the same side of AB, then A,B,P,Q are concyclic.
P Q
  ABQP is a cyclic quadrilateral
A B

9. Converse Theorems
(1) The circle whose diameter is the hypotenuse of a right angled
triangle passes through the third vertex.
C
ABC are concyclic with AB diameter
 in a semicircle  90 

A B
(2) If an interval AB subtends the same angle at two points P and Q on
the same side of AB, then A,B,P,Q are concyclic.
P Q
  ABQP is a cyclic quadrilateral
s in same segment are  
A B

10. (3) If a pair of opposite angles in a quadrilateral are supplementary (or
if an exterior angle equals the opposite interior angle) then the
quadrilateral is cyclic.

11. (3) If a pair of opposite angles in a quadrilateral are supplementary (or
if an exterior angle equals the opposite interior angle) then the
quadrilateral is cyclic.
D
120 C
60
A B

12. (3) If a pair of opposite angles in a quadrilateral are supplementary (or
if an exterior angle equals the opposite interior angle) then the
quadrilateral is cyclic.
D
120 C ABCD are concyclic
60
A B

13. (3) If a pair of opposite angles in a quadrilateral are supplementary (or
if an exterior angle equals the opposite interior angle) then the
quadrilateral is cyclic.
D
120 C ABCD are concyclic
A
60
B
opposites in cyclic quadrilateral are  180 


14. (3) If a pair of opposite angles in a quadrilateral are supplementary (or
if an exterior angle equals the opposite interior angle) then the
quadrilateral is cyclic.
D
120 C ABCD are concyclic
A
60
B
opposites in cyclic quadrilateral are  180 

Exercise 9D; 1, 2, 3, 6b, 7b, 10a, 11